Question

x - 3y = 6; y = (1/3)x - 2; which describes the system of equations above? consistent and dependent; consistent and independent; inconsistent

Explanation:

Step1: Rewrite the first equation

Start with the equation \( x - 3y = 6 \). Let's solve for \( y \) to get it in slope - intercept form (\( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept).
Subtract \( x \) from both sides: \( - 3y=-x + 6 \)
Divide each term by - 3: \( y=\frac{-x}{-3}+\frac{6}{-3} \), which simplifies to \( y = \frac{1}{3}x-2 \)

Step2: Compare the two equations

The second equation is already given as \( y=\frac{1}{3}x - 2 \). We can see that the two equations \( x - 3y = 6 \) (which we rewrote as \( y=\frac{1}{3}x-2 \)) and \( y=\frac{1}{3}x - 2 \) are identical.

For a system of linear equations:

• If the two equations represent the same line (same slope and same y - intercept), the system is consistent (has at least one solution) and dependent (infinitely many solutions, since all points on the line are solutions).
• If the equations represent two distinct intersecting lines (same slope, different y - intercepts), the system is consistent and independent (one solution).
• If the equations represent two parallel lines (same slope, different y - intercepts), the system is inconsistent (no solutions).

Since our two equations are the same line, the system is consistent (because there are solutions - all points on the line) and dependent (because the equations are dependent, representing the same line).

Answer:

consistent and dependent